How do I change Z offset in Blender 2.8?
You have 3 options;


Make whatever is in front of the object semi-transparent.
Render the object, then render what is in front of it (each perhaps on their own render layer), then composite the two renders in the compositer.
Move the object closer to the camera than the object that is obscuring it.


As far as I know, there is no "z-offset" setting for an object in blender.

Edit for followup comments

Given your example, I've come up with the following proof of concept;







I have attached the .blend file;



Explanation

First, forgive my model, it was a quick mock-up. By the time you sent your blend, I was almost ready to post :)

So, what I've done here is created a series of spheres, nested inside each other;



Then, I flipped them inside out (so to speak - I've actually just flipped the normal of all the faces by going into edit mode, selecting everything, searching for "flip normals", and clicking that option). This allows you to see all the spheres, despite being inside each other.

Note: This should be visible by default in Material Preview and Render modes, but if you want to see it in Solid mode, you'll need to turn on backface culling specifically for that mode;





Now, as for the shading, I set up the following node structure;



Effectively, what this does is flat shades the object (so you don't have the odd, inverted, shaded spheres).

To note, this is a default ColorRamp;



I only wanted the one color (no shading, no variation), so I completely deleted the first slider. Then I changed the color to the desired value.